As stated above, prior works have shown positive results when using score models to solve ill-posed inverse problems like sub-sampled image reconstruction [1]}, [2]}, [3]}. In prior work, however, the forward model is assumed to be fixed and known throughout the reconstruction procedure. In our work, however, we assume that our forward operator \(A_{\kappa }\)  belongs to a restricted class of operators with unknown parameters \(\kappa \)  which must be learned during the inference. Another way of viewing this problem is as posterior sampling from the joint distribution \(p(x,\kappa |y)\) . Under this joint posterior, we arrive at the following update equations:
\(x_{t+1} &= x_t + \eta _t\nabla _{x_t} \log {p(x_t,\kappa _t|y)} + z_t, \\\kappa _{t+1} &= \kappa _t + \lambda _t \nabla _{\kappa _t} \log {p(x_t,\kappa _t|y)} + \xi _t,\) 
